Capacitance

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Cornell University *

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2213

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Electrical Engineering

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Jan 9, 2024

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T E M P L E U N I V E R S I T Y P H Y S I C S Capacitance Capacitors store charge and perform many other useful functions in circuits. The charge Q on a capacitor’s plates is proportional to the potential difference V across the capacitor. Q = CV (1) where C is a proportionality constant known as the capacitance . Capacitance has units of farads (F) (1 farad = 1 coulomb/volt). In the first part of this lab, we will investigate the time dependence of capacitor charging and discharging. In the second part of the lab, we will look at parallel plate capacitors, one of the most common types of capacitors (the tiny capacitors we see on circuit boards are often very thin parallel plates that have been rolled up into a tube shape). We will examine how the separation of the plates affects the capacitance. Picture a charged capacitor as being a bunch of electrons on a piece of metal, then you can further imagine that when a wire is connected to the capacitor, the electrons will want to drain off of the capacitor - and what are moving electrons but current ? The capacitor only has a finite capacity for charge, so the current will decrease with time. This process causes the potential V across the capacitor to decrease exponentially with time as V ( t )= V 0 e − t RC (2) where V 0 is the initial potential across the capacitor, t is time, R is the resistance in the circuit and C is the capacitance. The rate of the decrease is determined by the product RC , known as the time constant of the circuit. When the capacitor is charged, the potential across it approaches the final value V 0 following the relation V ( t )= V 0 ( 1 − e − t RC ) (3) As you can see, the same time constant RC describes the rate of charging as well as the rate of discharging. For an analogy to this type of exponential decay, imagine a candy jar that is initially filled with 1000 candies and once an hour you eat 10 % of the candies in the jar. Clearly, the 10 % comprises a smaller and smaller number of candies each time. For an exercise, sketch a rough graph of the number of candies vs. time; how would the graph change if instead of 10 %, you removed 20 % at a time? Learning Goals for This Laboratory: 1

SPDT switch Voltage Probes Figure 1 T E M P L E U N I V E R S I T Y P H Y S I C S  Become familiar with exponential decay including the time constant and the effect of R and C values.  Understand how the plate separation affects the capacitance of a parallel plate capacitor.  Practice building a more advanced circuit based on a schematic. Apparatus: computer with Pasco 850 interface, Capstone and Excel software, DC power supply, current and voltage probes, wires, light bulb, 10 µF capacitor, various resistors, two D batteries, single-pole double-throw switch (SPDT) Part I. Capacitor Charging and Discharging 1. Build the circuit as shown in Figure 1 with the C = 10 µF capacitor and R = 100 k Ω resistor. The batteries should be used as the voltage source. Use the Switch SW2 on the circuit board as the switch. Also make sure SW1 on the circuit board is set to the “BATTERY” position to use the batteries as the power supply. 2. Attach the red positive lead of the voltage probe to the side of the capacitor connected to the resistor and connect the black lead to the other side of the capacitor. 3. Prepare the computer for data collection. Make an x-y graph, with voltage on the y-axis time on the x-axis. Data Collection We’ll first collect data on discharging when the 100 k Ω resistor is in the circuit. Then we look at charging. Next we switch to a smaller resistor and repeat: discharging, then charging. 1. Collect your data in Continuous Mode. Charge the capacitor for 30 s or so with the switch closed (exactly as it is illustrated in Figure 2). Now let’s collect data for discharging. First click Record and then throw the switch. Hit Stop once the voltage reaches 0. (Repeat if necessary to obtain clean data.) Capstone will automatically name this run Run#1 for you. Do not delete your runs. 2 2/3/2023 12:24 PM

T E M P L E U N I V E R S I T Y P H Y S I C S 2. The capacitor is now discharged. Let’s now record data while charging the capacitor. Everything is already set, so click Record and then throw the switch the other way. Allow the data collection to run until the capacitor reaches its max voltage (what is the max voltage?) and then hit Stop . Differences in the charging and discharging processes should be readily apparent when the two are displayed on the same graph. At the top of the graph, turn on the Select Data button (it is the multicolor triangle shaped button) and the use the drop-down menu to select which runs to display. Question 1. How do the shapes of the charging and discharging graphs compare? 3. Now repeat the experiment with a resistor of lower value. How do you think this change will affect the way the capacitor discharges (it might help to use the analogy of water current in place of electric current, with the resistor restricting the flow)? Switch to the 47 k Ω resistor and repeat data collection for charging and discharging as above. Data Analysis Now we will fit the 100k for discharging and charging to a curve function and compare this to the theoretical functions, Equations 2 and 3 above. 1. The basic process is to display the data you want to fit using the data display tool (multicolor triangle) and then highlight the region to fit using the highlighter tool. Then turn on the data fitting tool and choose the appropriate function in the drop-down menu. When highlighting the data in the graph, omit the constant portion that got recorded just before throwing the switch. Also try adjusting the size of your highlighted region to obtain a good fit. Generally, you don’t want the highlighted region to cover all your data; a smaller region provides a better fit if it has a large enough number of points within it. After you select a fit equation, check that the fit line does a good job of aligning to your data. If not, try resizing your highlighter box, and if that doesn’t work, you can edit the fit parameters manually. To manually edit the fit parameters, attempt to fit the data first, then double click on the fit equation to open the curve fit editor. Then you can uncheck any locked values and click reset. If that doesn’t improve the fit, you can calculate what the time constant should be and enter this value for the coefficient B and check the box to lock the value to force the fit to use this value. Click to update the fit and see how well it worked. Iterate this process as needed until you have a good fit to your data. 2. Fit both the charging and discharging data on the same graph and take a screenshot for your lab report. Also make a graph showing both the 47-k data and 100-k data to easily compare them and save a screenshot for your report. 3. Record the value of the fit parameters in an Excel table. 3 2/3/2023 12:24 PM

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T E M P L E U N I V E R S I T Y P H Y S I C S Question 2. Compare the fit function parameter B to the time constant RC in Equations 2 and 3. How are they related? 4. Given the resistor and capacitor values we used, calculate the expected time constant of the two circuits; make sure to use SI units correctly and consistently. Compare these to the actual time constants we measured by taking the inverse of the fit constant B for each trial. Question 3. How close were the actual (measured) time constants to what you expected given the nominal values of resistance and capacitance (calculate percent difference)? Question 4. Look at your graphs for 47k vs. those for 100k. Qualitatively speaking, what effect did decreasing the resistance have on capacitor discharging/ charging times? Question 5. For the 100k data, did your charging and discharging have about the same time constant? Should they have the same time constant? Question 6. How would the graphs of your discharge data look if, instead of plotting the potential vs. time, you plotted the natural logarithm of the potential vs. time? Sketch a prediction. Now plot it by selecting your ‘discharging 100k’ data and hiding the other runs. Highlight the discharging part of the run and then click on the y-axis label and select ln(V). What is the significance of the slope of the plot of ln(V) vs. time for a capacitor discharge circuit? Hint: take the natural log of both sides of Equation 2. Question 7. What percentage of the initial potential remains after one time constant has passed? After two time constants? Three? 5. Calculate the energy (in Joules) stored in the capacitor when it is fully charged using the equation. Look in your text for the equation. Part II. Capacitance vs. Plate Separation The capacitance C is constant for a given capacitor. For a parallel plate capacitor, its value probably depends on the area A and distance d between the plates. We will place a fixed amount of charge on the parallel plate capacitor and investigate how the voltage varies with the separation. This investigation will allow you to establish how the capacitance varies with the separation between the plates . Question 8. Our plan is to place a fixed amount of charge on the plates and then measure the voltage on the plates as we vary their separation; how does measuring the voltage across the plates give us information about the capacitance ? 1. Prepare a data table with columns for voltage and plate separation distance. 4 2/3/2023 12:24 PM

T E M P L E U N I V E R S I T Y P H Y S I C S 2. Set up the equipment as shown in the diagram with the parallel plate capacitor connected to the electrometer via the locking BNC connector. It’s essential the electrometer and voltage source be grounded to earth; a ground wire connected to the building is provided. Connect leads to the 30-V output and ground of the voltage source, but don’t connect them to the plates of the capacitor. 3. Now charge the capacitor once by touching the positive voltage source lead to one plate and the negative lead to the other plate then removing the leads. This way, the charge is fixed on the plates because they are isolated from the power supply so that no charge can be added or removed from this point forward. Remove the leads and record the electrometer reading in your data table. Note: the plates can be discharged at any time by shorting them together with any conductor 4. Now, by only touching the plastic base of the plates, record the electrometer reading as you vary the separation distance in 1 mm increments, noting that the optimal starting separation distance and electrometer range setting may need to be adjusted as environmental conditions vary. The following values are often suitable. Modify as needed to collect data over a range of distance values. Initial plate separation = 0.5 cm. Electrometer range = 10 Volts full scale. When recording data, the person moving the plates should keep their hands on the plates for the duration to avoid transient effects caused by moving your hands on and off the plates. 5. Using Excel, make a coordinate (scatter) plot of the voltage vs. separation distance. Question 9. What does the graph of voltage vs. plate separation look like? How does the potential vary with separation? Question 10. Based on the results of the investigation of a parallel-plate capacitor, how does its capacitance vary with the separation distance d ? 5 2/3/2023 12:24 PM Voltage source leads. Don’t connect these to the plates. Ground wire electrometer leads for measuring V Ground wire to building

Capacitance

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